Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [40,5,Mod(17,40)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("40.17"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 40.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.13479852335\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.313431616.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 27x^{4} + 145x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 33.3
Root \(-4.49029i\) of defining polynomial
Character \(\chi\) \(=\) 40.33
Dual form 40.5.l.c.17.3

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(8.28874 + 8.28874i) q^{3} +(-10.2887 + 22.7847i) q^{5} +(-3.20721 + 3.20721i) q^{7} +56.4063i q^{9} -19.5694 q^{11} +(219.840 + 219.840i) q^{13} +(-274.137 + 103.576i) q^{15} +(281.352 - 281.352i) q^{17} -305.648i q^{19} -53.1675 q^{21} +(-633.243 - 633.243i) q^{23} +(-413.284 - 468.851i) q^{25} +(203.851 - 203.851i) q^{27} +885.210i q^{29} +1383.32 q^{31} +(-162.205 - 162.205i) q^{33} +(-40.0772 - 106.073i) q^{35} +(453.987 - 453.987i) q^{37} +3644.39i q^{39} -193.003 q^{41} +(-1016.66 - 1016.66i) q^{43} +(-1285.20 - 580.350i) q^{45} +(-1203.53 + 1203.53i) q^{47} +2380.43i q^{49} +4664.11 q^{51} +(-530.174 - 530.174i) q^{53} +(201.344 - 445.882i) q^{55} +(2533.43 - 2533.43i) q^{57} +934.801i q^{59} -1122.32 q^{61} +(-180.907 - 180.907i) q^{63} +(-7270.86 + 2747.11i) q^{65} +(4760.40 - 4760.40i) q^{67} -10497.6i q^{69} -8831.10 q^{71} +(3008.63 + 3008.63i) q^{73} +(460.584 - 7311.79i) q^{75} +(62.7631 - 62.7631i) q^{77} +3349.09i q^{79} +7948.24 q^{81} +(-3166.08 - 3166.08i) q^{83} +(3515.76 + 9305.28i) q^{85} +(-7337.27 + 7337.27i) q^{87} +1732.33i q^{89} -1410.15 q^{91} +(11466.0 + 11466.0i) q^{93} +(6964.09 + 3144.73i) q^{95} +(1659.43 - 1659.43i) q^{97} -1103.84i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{3} - 4 q^{5} - 40 q^{7} + 72 q^{11} - 62 q^{13} - 280 q^{15} + 418 q^{17} + 1736 q^{21} - 1760 q^{23} - 2974 q^{25} + 2248 q^{27} + 2864 q^{31} - 8 q^{33} - 4224 q^{35} + 5366 q^{37} + 1656 q^{41}+ \cdots - 9418 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 8.28874 + 8.28874i 0.920971 + 0.920971i 0.997098 − 0.0761273i \(-0.0242555\pi\)
−0.0761273 + 0.997098i \(0.524256\pi\)
\(4\) 0 0
\(5\) −10.2887 + 22.7847i −0.411549 + 0.911387i
\(6\) 0 0
\(7\) −3.20721 + 3.20721i −0.0654533 + 0.0654533i −0.739076 − 0.673622i \(-0.764737\pi\)
0.673622 + 0.739076i \(0.264737\pi\)
\(8\) 0 0
\(9\) 56.4063i 0.696374i
\(10\) 0 0
\(11\) −19.5694 −0.161730 −0.0808652 − 0.996725i \(-0.525768\pi\)
−0.0808652 + 0.996725i \(0.525768\pi\)
\(12\) 0 0
\(13\) 219.840 + 219.840i 1.30083 + 1.30083i 0.927833 + 0.372996i \(0.121669\pi\)
0.372996 + 0.927833i \(0.378331\pi\)
\(14\) 0 0
\(15\) −274.137 + 103.576i −1.21839 + 0.460336i
\(16\) 0 0
\(17\) 281.352 − 281.352i 0.973537 − 0.973537i −0.0261217 − 0.999659i \(-0.508316\pi\)
0.999659 + 0.0261217i \(0.00831574\pi\)
\(18\) 0 0
\(19\) − 305.648i − 0.846670i −0.905973 − 0.423335i \(-0.860859\pi\)
0.905973 − 0.423335i \(-0.139141\pi\)
\(20\) 0 0
\(21\) −53.1675 −0.120561
\(22\) 0 0
\(23\) −633.243 − 633.243i −1.19706 − 1.19706i −0.975044 − 0.222014i \(-0.928737\pi\)
−0.222014 − 0.975044i \(-0.571263\pi\)
\(24\) 0 0
\(25\) −413.284 − 468.851i −0.661254 − 0.750162i
\(26\) 0 0
\(27\) 203.851 − 203.851i 0.279630 − 0.279630i
\(28\) 0 0
\(29\) 885.210i 1.05257i 0.850309 + 0.526284i \(0.176415\pi\)
−0.850309 + 0.526284i \(0.823585\pi\)
\(30\) 0 0
\(31\) 1383.32 1.43946 0.719730 − 0.694254i \(-0.244266\pi\)
0.719730 + 0.694254i \(0.244266\pi\)
\(32\) 0 0
\(33\) −162.205 − 162.205i −0.148949 − 0.148949i
\(34\) 0 0
\(35\) −40.0772 − 106.073i −0.0327160 − 0.0865906i
\(36\) 0 0
\(37\) 453.987 − 453.987i 0.331619 − 0.331619i −0.521582 − 0.853201i \(-0.674658\pi\)
0.853201 + 0.521582i \(0.174658\pi\)
\(38\) 0 0
\(39\) 3644.39i 2.39605i
\(40\) 0 0
\(41\) −193.003 −0.114814 −0.0574071 − 0.998351i \(-0.518283\pi\)
−0.0574071 + 0.998351i \(0.518283\pi\)
\(42\) 0 0
\(43\) −1016.66 − 1016.66i −0.549846 − 0.549846i 0.376550 − 0.926396i \(-0.377110\pi\)
−0.926396 + 0.376550i \(0.877110\pi\)
\(44\) 0 0
\(45\) −1285.20 − 580.350i −0.634667 − 0.286592i
\(46\) 0 0
\(47\) −1203.53 + 1203.53i −0.544828 + 0.544828i −0.924940 − 0.380112i \(-0.875885\pi\)
0.380112 + 0.924940i \(0.375885\pi\)
\(48\) 0 0
\(49\) 2380.43i 0.991432i
\(50\) 0 0
\(51\) 4664.11 1.79320
\(52\) 0 0
\(53\) −530.174 − 530.174i −0.188741 − 0.188741i 0.606410 − 0.795152i \(-0.292609\pi\)
−0.795152 + 0.606410i \(0.792609\pi\)
\(54\) 0 0
\(55\) 201.344 − 445.882i 0.0665600 − 0.147399i
\(56\) 0 0
\(57\) 2533.43 − 2533.43i 0.779758 − 0.779758i
\(58\) 0 0
\(59\) 934.801i 0.268544i 0.990945 + 0.134272i \(0.0428695\pi\)
−0.990945 + 0.134272i \(0.957130\pi\)
\(60\) 0 0
\(61\) −1122.32 −0.301617 −0.150809 − 0.988563i \(-0.548188\pi\)
−0.150809 + 0.988563i \(0.548188\pi\)
\(62\) 0 0
\(63\) −180.907 − 180.907i −0.0455800 − 0.0455800i
\(64\) 0 0
\(65\) −7270.86 + 2747.11i −1.72091 + 0.650204i
\(66\) 0 0
\(67\) 4760.40 − 4760.40i 1.06046 − 1.06046i 0.0624093 − 0.998051i \(-0.480122\pi\)
0.998051 − 0.0624093i \(-0.0198784\pi\)
\(68\) 0 0
\(69\) − 10497.6i − 2.20491i
\(70\) 0 0
\(71\) −8831.10 −1.75185 −0.875927 − 0.482443i \(-0.839750\pi\)
−0.875927 + 0.482443i \(0.839750\pi\)
\(72\) 0 0
\(73\) 3008.63 + 3008.63i 0.564578 + 0.564578i 0.930604 − 0.366027i \(-0.119282\pi\)
−0.366027 + 0.930604i \(0.619282\pi\)
\(74\) 0 0
\(75\) 460.584 − 7311.79i 0.0818816 − 1.29987i
\(76\) 0 0
\(77\) 62.7631 − 62.7631i 0.0105858 − 0.0105858i
\(78\) 0 0
\(79\) 3349.09i 0.536627i 0.963332 + 0.268313i \(0.0864663\pi\)
−0.963332 + 0.268313i \(0.913534\pi\)
\(80\) 0 0
\(81\) 7948.24 1.21144
\(82\) 0 0
\(83\) −3166.08 − 3166.08i −0.459586 − 0.459586i 0.438934 − 0.898519i \(-0.355356\pi\)
−0.898519 + 0.438934i \(0.855356\pi\)
\(84\) 0 0
\(85\) 3515.76 + 9305.28i 0.486611 + 1.28793i
\(86\) 0 0
\(87\) −7337.27 + 7337.27i −0.969384 + 0.969384i
\(88\) 0 0
\(89\) 1732.33i 0.218700i 0.994003 + 0.109350i \(0.0348770\pi\)
−0.994003 + 0.109350i \(0.965123\pi\)
\(90\) 0 0
\(91\) −1410.15 −0.170287
\(92\) 0 0
\(93\) 11466.0 + 11466.0i 1.32570 + 1.32570i
\(94\) 0 0
\(95\) 6964.09 + 3144.73i 0.771644 + 0.348446i
\(96\) 0 0
\(97\) 1659.43 − 1659.43i 0.176366 − 0.176366i −0.613404 − 0.789770i \(-0.710200\pi\)
0.789770 + 0.613404i \(0.210200\pi\)
\(98\) 0 0
\(99\) − 1103.84i − 0.112625i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 40.5.l.c.33.3 yes 6
3.2 odd 2 360.5.v.c.73.3 6
4.3 odd 2 80.5.p.g.33.1 6
5.2 odd 4 inner 40.5.l.c.17.3 ✓ 6
5.3 odd 4 200.5.l.e.57.1 6
5.4 even 2 200.5.l.e.193.1 6
8.3 odd 2 320.5.p.o.193.3 6
8.5 even 2 320.5.p.p.193.1 6
15.2 even 4 360.5.v.c.217.3 6
20.3 even 4 400.5.p.o.257.3 6
20.7 even 4 80.5.p.g.17.1 6
20.19 odd 2 400.5.p.o.193.3 6
40.27 even 4 320.5.p.o.257.3 6
40.37 odd 4 320.5.p.p.257.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.c.17.3 ✓ 6 5.2 odd 4 inner
40.5.l.c.33.3 yes 6 1.1 even 1 trivial
80.5.p.g.17.1 6 20.7 even 4
80.5.p.g.33.1 6 4.3 odd 2
200.5.l.e.57.1 6 5.3 odd 4
200.5.l.e.193.1 6 5.4 even 2
320.5.p.o.193.3 6 8.3 odd 2
320.5.p.o.257.3 6 40.27 even 4
320.5.p.p.193.1 6 8.5 even 2
320.5.p.p.257.1 6 40.37 odd 4
360.5.v.c.73.3 6 3.2 odd 2
360.5.v.c.217.3 6 15.2 even 4
400.5.p.o.193.3 6 20.19 odd 2
400.5.p.o.257.3 6 20.3 even 4